### Browsing by Author "Fredericks, Ebrahim"

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- ItemOpen AccessClassification, Valuation and Real Options Analysis of Climate Change Projects in Africa: A case study of Ghana in West Africa(2022) Asumadagwine, Godwin; Fredericks, Ebrahim; Becker, Ronnie; Ikpe, DennisProjects and investments such as those of R & D and climate change are subject to several uncertainties. These uncertainties, if not properly managed, could defeat the actual purpose and target of the project. In this work, we identify flexibility as a way in which uncertainties can be managed. As the main aim of investments is to make profit, it is significant that investors conduct an in depth study of the project under consideration. This will aid in cost-benefit analysis to ascertain whether it is financially worth it or not. Real options in finance, is the tool that has proved effective in that regard. However, not every project can be analysed using real options. This thesis introduces real options in climate change investment in Africa (Ghana), 2014-2020. To determine whether real options could be applied, we introduce and estimate certain measures: flexibility, optionability and realizability. These metrics help us to identify the project in which real options can be used. In this case, we characterize real options into mechanisms and types. The mechanisms are noted to be the enablers of real options while the types are the particular ones enabled. This work also introduces the Decision Structure Matrix (DSM) in climate change investment. The logical Coupled Dependency Matrix (C-DSM) is used to specify the logical relations that exist among dependencies in the project. The logical dependencies are then used for the estimation of the metrics . This study serves as the basis for the application of real options analysis in climate change investments.
- ItemOpen AccessDiscrete symmetry analysis of partial differential equations for bond pricing(2018) Ledwaba, Nomsa Maripa; Fredericks, EbrahimWe show how to compute the discrete symmetries for a given Black-Scholes (B-S) partial differential equation (PDE) with the aid of the full automorphism group of the Lie algebra associated to the standard B-S PDE. The paper determines the discrete symmetries using two methods. The first is by G. Silberberg which determines the full automorphism group by constructing the symmetry generators' centralizer and Lie algebra's radical. The other is by P. Hydon which is based on the observation that the adjoint action of any point symmetry of a partial differential equation is an automorphism of the PDE's Lie point symmetry algebra [27]. Automorphisms are essential for constructing discrete symmetries of a given partial differential equation. How does one _t in this mathematical concept in the application of finance? The concept of arbitrage which in certain circumstances allows us to establish the precise relationship between prices and thence how to determine prices, underlies the theory of financial derivatives pricing and hedging [40]. We use arbitrage together with the Black-Scholes model for asset price movements when trading derivative securities. 1Arbitrage is used to creating a portfolio and the discrete symmetries show how to create a portfolio. Gazizov and Ibragimov [10], computed the Lie point symmetries of the Black-Scholes PDE and found an infinite dimensional Lie algebra of infinitesimal symmetries generated by the operators. Discrete symmetries are more effective on PDEs since they are not held back by boundary conditions and are used in1. equivalent bifurcation theory; 2. construction of invariant solutions; 3. simplification of numerical schemes. 4. used in put-call parity relationship (see application in finance); 5. used in put-call symmetry relationship (see application in finance)
- ItemOpen AccessLie Analysis for Partial Differential Equations in Finance(2019) Nhangumbe, Clarinda Vitorino; Fredericks, Ebrahim; Canhanga , BetuelWeather derivatives are financial tools used to manage the risks related to changes in the weather and are priced considering weather variables such as rainfall, temperature, humidity and wind as the underlying asset. Some recent researches suggest to model the amount of rainfall by considering the mean reverting processes. As an example, the Ornstein Uhlenbeck process was proposed by Allen [3] to model yearly rainfall and by Unami et al. [52] to model the irregularity of rainfall intensity as well as duration of dry spells. By using the Feynman-Kac theorem and the rainfall indexes we derive the partial differential equations (PDEs) that governs the price of an European option. We apply the Lie analysis theory to solve the PDEs, we provide the group classification and use it to find the invariant analytical solutions, particularly the ones compatible with the terminal conditions.
- ItemOpen AccessModelling credit spreads in an illiquid South African corporate debt market(2019) Jones, Samantha; Laurie, Henri; Fredericks, Ebrahim; Becker, Ronald; Dugmore, BrettThe South African debt market suffers from severe illiquidity, as is common in most emerging markets. Infrequent trading leads to out-of-date market prices and stale, unreliable credit spreads. Since the coverage of the South African debt market by credit ratings agencies is poor, meaningful credit spreads become even more important in gauging credit worth. The illiquidity of corporate vanilla bonds traded on the Johannesburg Stock Exchange and the ensuing adverse effects on their credit spreads are rigourously illustrated. Lack of data poses a serious problem when modelling any system and this analysis provides motivation for the necessity of a framework that addresses the statistical complications that incomplete data sets present. A new model, which is a distinctive modification of the well-known mean-reverting Ornstein-Uhlenbeck or Vasicek process, is introduced. This innovative approach creates a mathematically and intuitively sound relationship between the credit spread process and that of the stock price of the bond issuer. This key feature is used in a Bayesian methodology to impute missing credit spread data for calibration, for more meaningful inference. On sparse simulated data and market observed credit spread time series, the model proves to deliver an improved quality of the estimations, with probabilities that are now statistically founded. Even on complete credit spread time series, the model is shown to have some merit over the traditional model in terms of goodness of fit, giving further credence to its validity and explanatory power.
- ItemOpen AccessPoint symmetry methods for Itô Stochastic Differential Equations (SDE) with a finite jump process(2017) Nass, Aminu Ma'aruf; Fredericks, EbrahimThe mixture of Wiener and a Poisson processes are the primary tools used in creating jump-diffusion process which is very popular in mathematical modeling. In financial mathematics, they are used to describe the change of stock rates and bonanzas, and they are often used in mathematical biology modeling and population dynamics. In this thesis, we extended the Lie point symmetry theory of deterministic differential equations to the class of jump-diffusion stochastic differential equations, i.e., a stochastic process driven by both Wiener and Poisson processes. The Poisson process generates the jumps whereas the Brownian motion path is continuous. The determining equations for a stochastic differential equation with finite jump are successfully derived in an Itô calculus context and are found to be deterministic, even though they represent a stochastic process. This work leads to an understanding of the random time change formulae for Poisson driven process in the context of Lie point symmetries without having to consult much of the intense Itô calculus theory needed to formally derive it. We apply the invariance methodology of Lie point transformation together with the more generalized Itô formulae, without enforcing any conditions to the moments of the stochastic processes to derive the determining equations and apply it to few models. In the first part of the thesis, point symmetry of Poisson-driven stochastic differential equations is discussed, by considering the infinitesimals of not only spatial and temporal variables but also infinitesimals of the Poisson process variable. This was later extended, in the second part, to define the symmetry of jumpdiffusion stochastic differential equations (i.e., stochastic differential equations driven by both Wiener and Poisson processes).
- ItemOpen AccessThe required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation(2011) Masike, Kakanyo Knowledge; Fredericks, Ebrahim; Ebobisse Bille, FrancoisIn this thesis we demonstrate how to obtain the required ansatz to determine Lie point transformations of evolution-type equations from the contact transformation approach. We indicate that the Lie point transformations of the Fokker-Planck equation (FPE), which is a second-order linear parabolic partial differential equation (PDE), are projectable by using the ansatz. We further obtain the symmetries of a stochastic ordinary differential equation (SODE) which corresponds to those of the FPE. This is possible because there exists a relationship between an SODE and the associated (deterministic) FPE. The study of SODEs is an interesting and applicable concept in the real world and one of the building factors to this study is an Ito integral. These Ito integrals are of much use, for instance, in the field of mathematical finance whereby its use has shown the relationship between call options and their non-deterministic underlying stock prices. Wiener processes must be considered in finding an approximation of these integrals. Acclimatization of Sophus Lie's work to SODEs has been done by (Gaeta and Quintero [2]; Wafo Soh and Mahomed [41]; Unal [42]; Fredericks and Mahomed [43]). The determining equations for the first-order SODEs are derived in an Ito calculus context and are non-stochastic. Consequently, symmetries of an SODE are obtained without the consultation of its corresponding FPE.